



01627nas a2200193   4500008004100000245002900041210002900070260001400099300001500113490000800128520112700136100002801263700002301291700002301314700001901337700002001356700002001376856003701396       2020                        eng d00aQuantum Coupon Collector0 aQuantum Coupon Collector  c2/18/2020  a10:1-10:170 v1583 a<p>We study how efficiently a k-element set S&sube;[n] can be learned from a uniform superposition |S&rang; of its elements. One can think of |S&rang;=&sum;i&isin;S|i&rang;/|S|&minus;&minus;&minus;&radic; as the quantum version of a uniformly random sample over S, as in the classical analysis of the ``coupon collector problem.&#39;&#39; We show that if k is close to n, then we can learn S using asymptotically fewer quantum samples than random samples. In particular, if there are n&minus;k=O(1) missing elements then O(k) copies of |S&rang; suffice, in contrast to the Θ(klogk) random samples needed by a classical coupon collector. On the other hand, if n&minus;k=Ω(k), then Ω(klogk) quantum samples are~necessary. More generally, we give tight bounds on the number of quantum samples needed for every k and n, and we give efficient quantum learning algorithms. We also give tight bounds in the model where we can additionally reflect through |S&rang;. Finally, we relate coupon collection to a known example separating proper and improper PAC learning that turns out to show no separation in the quantum case.</p>1 aArunachalam, Srinivasan1 aBelovs, Aleksandrs1 aChilds, Andrew, M.1 aKothari, Robin1 aRosmanis, Ansis1 ade Wolf, Ronald  uhttps://arxiv.org/abs/2002.0768802244nas a2200133   4500008004100000245006000041210006000101260001500161520182800176100002802004700002002032700002102052856003702073       2019                        eng d00aQuantum hardness of learning shallow classical circuits0 aQuantum hardness of learning shallow classical circuits  c03/07/20193 a<p>In this paper we study the quantum learnability of constant-depth classical circuits under the uniform distribution and in the distribution-independent framework of PAC learning. In order to attain our results, we establish connections between quantum learning and quantum-secure cryptosystems. We then achieve the following results. 1) Hardness of learning AC0 and TC0 under the uniform distribution. Our first result concerns the concept class TC0 (resp. AC0), the class of constant-depth and polynomial-sized circuits with unbounded fan-in majority gates (resp. AND, OR, NOT gates). We show that if there exists no quantum polynomial-time (resp. sub-exponential time) algorithm to solve the Learning with Errors (LWE) problem, then there exists no polynomial-time quantum learning algorithm for TC0 (resp. AC0) under the uniform distribution (even with access to quantum membership queries). The main technique in this result uses explicit pseudo-random generators that are believed to be quantum-secure to construct concept classes that are hard to learn quantumly under the uniform distribution. 2) Hardness of learning TC02 in the PAC setting. Our second result shows that if there exists no quantum polynomial time algorithm for the LWE problem, then there exists no polynomial time quantum PAC learning algorithm for the class TC02, i.e., depth-2 TC0 circuits. The main technique in this result is to establish a connection between the quantum security of public-key cryptosystems and the learnability of a concept class that consists of decryption functions of the cryptosystem. This gives a strong conditional negative answer to one of the &quot;Ten Semi-Grand Challenges for Quantum Computing Theory&quot; raised by Aaronson [Aar05], who asked if AC0 and TC0 can be PAC-learned in quantum polynomial time.</p>1 aArunachalam, Srinivasan1 aGrilo, Alex, B.1 aSundaram, Aarthi  uhttps://arxiv.org/abs/1903.02840